goodInitialPair(List) -- make an initial pair conjectured to be good by Shub and Smale

Synopsis

Function: goodInitialPair
Usage:

(S,sol) = goodInitialPair T
Inputs:
- T, a list, contains homogeneous polynomials
Optional inputs:
- GeneralPosition => ..., default value false, make a random unitary change of coordinates
Outputs:
- S, contains homogeneous polynomials
- sol, contains one solution of S

Description

Generates a start system S that is conjectured to have good complexity when used in linear homotopy with target system T leading to one solution. For more details see: C. Beltran and A. Leykin, "Certified numerical homotopy tracking", Experimental Mathematics 21(1): 69-83 (2012)

i1 : T = randomSd {2,3};

i2 : (S,solsS) = goodInitialPair T

                                 2
o2 = ({1.41421x1*x3, 1.73205x2*x3 }, {{0, 0, 1}})

o2 : Sequence

i3 : M = track(S,T,solsS,gamma=>0.6+0.8*ii,Software=>M2)

o3 = {{.751732+.463283*ii, .284351-.0717164*ii, .348144-.114302*ii}}

o3 : List

Ways to use this method: